dV.. 



dx 



dx 



dx 



de. 



dz 









Vx 



^x Yg 



1=0 





EA 



^e 



1 = 







lyy^y 



+ lyzM. 





E(I,, 



Izz- 



I..^) 



\A 



+ Izz 



K 





E(I I -I ^) 



*■ yy zz yz ' 



[10] 



Equations [10] summarize the elastic flexibility parameters of the beam, which describe 

 bending and tensile deformations in terms of bending moments and tensile force. 



SHEAR AND TORSION 



Next it is desired to determine the elastic flexibility parameters of the beam which 

 relate shear and torsion deformations to beam shear forces and twisting moment. We will use 

 Castigliano's Theorem, which requires that the total strain energy be expressed in terms of 

 the beam shear forces and twisting moment V , V , and M.. All the strain energy associated 

 with these deformations is in the shear of the plates. By statics it is shown that any single 

 plate sustains a shear flow q (force per unit length) which is the same at all points of the 

 plate (see page 54). Thus the shear strain energy of any plate j depends only on the shear flow 

 of that plate q.. The first step is to express all the shear flows q. as functions of V , V , 

 and M^. The second step is to express strain energy per unit length of the beam W as a 

 function of V , V^, and M^. The final step is to apply Castigliano's Theorem by differentiat- 

 ing this expression for W with respect to V , V , and M . 



To find the panel shear flows q. as functions of V , V , and M , we will first compose 

 q. of the sum of shears flowing in a tree q ^^^ ., a particular solution, and shears flowing in 

 loops q, 



The tree is selected by omitting sufficient plates so that the remaining plates are 

 simply connected. In a tree thus formed, all nodes of the section are part of the tree, and 

 any two nodes are connected by one and only one path through the tree. One node is arbi- 

 trairly selected as the "root" of the tree. 



Each loop is formed by taking one of the plates which was omitted in forming the tree 

 and all of the plates which are part of the tree. One loop is formed by this procedure, and 

 the plates of the tree which are part of this loop are retained in the loop, whereas extraneous 

 plates of the tree (not needed in the loop) are omitted from a description of the loop. 



The shear flows of the particular solution arise from the shear flowing out of the nodes 

 lout i- ^" ^ particular panel j, this shear flow q ^^^ . is given by the sum of q ^^^ . for all nodes 



64 



